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Rust claims to advance industrial programming by bridging the gap between low-level systems programming and high-level application programming. At the heart of the argument that this enables programmers to build more reliable and efficient software is the borrow checker - a novel approach to ownership that aims to balance type system expressivity with usability. And yet, to date there is no core type system that captures Rust's notion of ownership and borrowing, and hence no foundation for research on Rust to build upon. In this work, we set out to capture the essence of this model of ownership by developing a type systems account of Rust's borrow checker. We present Oxide, a formalized programming language close to source-level Rust (but with fully-annotated types). This presentation takes a new view of lifetimes as an approximation of the provenances of references, and our type system is able to automatically compute this information through a substructural typing judgment. We provide the first syntactic proof of type safety for borrow checking using progress and preservation. Oxide is a simpler formulation of borrow checking - including recent features such as non-lexical lifetimes - that we hope researchers will be able to use as the basis for work on Rust.